Derivation of nonlinear Gibbs measures from many-body quantum mechanics
Derivation of nonlinear Gibbs measures from many-body quantum mechanics
复制标题
多体量子力学非线性吉布斯测度的推导
DOI：
10.5802/jep.18
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
N. Rougerie
中科院分区：
文献类型：
--
作者：
Mathieu Lewin;P. T. Nam;N. Rougerie
我们证明，在温度 T 发散且相互作用表现为 1/T 的平均场极限下，可以从相应的多体、大正则、量子吉布斯态获得非线性吉布斯测度。我们继续将相互作用的吉布斯态描述为相对于非相互作用情况最小化计算自由能的函数。然后，我们利用量子相对熵的精细特性、相干态基础上的量子 de Finetti 测量和上/下符号之间的联系以及 Berezin-Lieb 型不等式，对相空间半经典分析进行无限维模拟。我们的结果涵盖了基于有限区间上的散焦非线性薛定谔函数的测量，以及维度 d\textgreater{}1 中更平滑的交互。
We prove that nonlinear Gibbs measures can be obtained from the corresponding many-body, grand-canonical, quantum Gibbs states, in a mean-field limit where the temperature T diverges and the interaction behaves as 1/T. We proceed by characterizing the interacting Gibbs state as minimizing a functional counting the free-energy relatively to the non-interacting case. We then perform an infinite-dimensional analogue of phase-space semiclassical analysis, using fine properties of the quantum relative entropy, the link between quantum de Finetti measures and upper/lower symbols in a coherent state basis, as well as Berezin-Lieb type inequalities. Our results cover the measure built on the defocusing nonlinear Schr{o}dinger functional on a finite interval, as well as smoother interactions in dimensions d\textgreater{}1.